Carl Schachter's article "Schoenberg's Hat" has the following sections:
Introduction: Schoenberg's Hat (327)Ascending and Descending Motion Contrasted (328)Structure and Contour (331)The Rising Fourth (333)Conclusion: Lewis Carroll's Trousers (337)Appendix: The Rising Urlinie (338)
Schachter begins by criticizing Schoenberg's extension of his idea of a unitary musical space (essential to derive and justify the several forms of a 12-tone row, including inversions) to tonal music, where Schachter points to "a functional context in which the tonal motive assumes…meaning…characterised by fixed points of orientation (the tonic note, triadic roots), [and] by important functional differences between rising and falling movement (for instance in the resolution of dissonances)" (328). Schachter is careful to say that notions of high, low, and others relating to space and motion are metaphorical or analogical, but then he goes on to claim that "I do not regard the analogies between musical space and motion on the one hand and physical space and motion on the other as unimportant, inexact though they may be. A significant part of music's ability to reflect our physical and emotional lives comes from just these analogies."
He then spends considerable time exploring text painting in relation to prolongational figures in songs by Schumann ("Die Lotosblume") and Brahms ("Der Tod, das ist die kühle Nacht"). Under Structure and Contour, the examples are Verdi, from "0 Terra, Addio" from the final duet of Aida; Schubert's "Die junge Nonne"; under the Rising Fourth, Bizet, Carmen's soliloquy in the "Card Trio" of the opera's third act. In the latter section, he also looks at two instrumental compositions: Bach's C-minor Fugue from WTC 1; Chopin's Prelude in E-major. The appendix comments on what Schachter regards as problems in my rising line article and renders his own judgments about analyses that are convincing with rising line Urlinien.
I will make some comments on the analyses and offer responses to the arguments in posts over the next several days.
References:
Schachter, Carl. "Schoenberg's Hat and Lewis Carroll's Trousers: Upward and Downward Motion in Musical Space." In Brenton Broadstock, Naomi Cumming, Denise Erdonmez Grocke, Catherine Falk, Ros McMillan, Kerry Murphy, Suzanne Robinson, and John Stinson, eds. Aflame with Music: 100 Years of Music at the University of Melbourne. Parkville, Victoria: Centre for Studies in Australian Music, 327-41.
Neumeyer, David. "The Ascending Urlinie." Journal of Music Theory 31/2 (1987): 275-303.
